Data Scaling

The wide variety of scales in astrophysical variables is a chronic problem for numerical simulation codes. When variables of widely disparate scales are combined in a machine calculation, the results are vulnerable to ​truncation error, especially if the order-of-magnitude difference between two values is below ​machine precision. Handled improperly, these scaling differences can cause massive problems in a simulation.

AstroBEAR handles the problem by dividing a scaling constant out of certain variables. The equations of ideal MHD are scale-free, so the scaling of variables does not change the results of the simulation. This does not hold true if physical source terms such as cooling or gravity are present; during source term steps, physical scaling must be reintroduced.

Quantities divided by a scaling term are sometimes referred to as scaled quantities. These scaled quantities are dimensionless, being described in terms of computational units.

Scaling in AstroBEAR

Data File Parameters

The most basic physical scaling options are set by the user in the physics.data file:

variable name	scale name (units)
nScale	number density scale (cm-3)
rScale	density scale (g cm-3)
pScale	pressure scale (dynes cm-2)
TempScale	temperature scale (K)
lScale	length scale (cm)

AstroBEAR derives other scaling constants such as magnetic pressure and velocity from these basic parameters:

variable name	scale name (units)	formula
VelScale	velocity scale (cm s-1)	\sqrt{\gamma ~p_{scale} \over r_{scale}}
BScale	magnetic field scale (esu)	\sqrt{p_{scale} \over {4 \pi}}
TimeScale	time scale (s)	l_{scale} \over v_{scale}
ScaleGrav*	gravity scale (cm3)	G ~r_{scale} ~{t_{scale}}^2
ScaleCool**	cooling scale (erg-1 cm3)	t_{scale} \over p_{scale}

*requires elliptic source terms.
**requires cooling source terms.

Example of Scaling Calculations

As an example, we will construct the scaling terms for the MHD clump simulations. We start with the following physical parameters for the simulation:

ambient density	100 cm-3
ambient temperature	104 K
wind speed	Mach 25
clump radius	200 AU
plasma beta	1

The first thing we notice is that the ambient wind speed is expressed as a ​Mach number instead of in cm/s. In essence, this quantity is pre-scaled to the sound speed of the problem, and does not need to be scaled any further. Similarly, the ​plasma beta term is a dimensionless quantity that specifies the strength of the magnetic field in terms of the pressure, so there is no need to concern ourselves with the scaling of the magnetic field.

The quantities that do need scaling are density, temperature, and distance.Since the clump radius is on the order of 100 AU, that seems like a reasonable value for lScale. AstroBEAR assumes that distance scales are given in centimeters, and 1 AU = 1.495 x 10¹³ cm; therefore, the scaling term in physics.data is:

lScale = 1.49598d15,

This gives us a scaled clump radius of 2 computational units.

Ideally, we would like to scale the ambient density to 1. Since we are given the number density but not the mass density, the density scales in physics.data should be set as follows:

nScale = 1.d2,
rScale = 0.d0,

This causes rScale to be derived from nScale, the known quantity.

Similarly, we would like to scale the ambient temperature to 1. To do this, we set the temperature and pressure scaling options in physics.data as follows:

pScale = 0.d0,
TempScale = 1.d4,

With this configuration, AstroBEAR both scales the ambient temperature to 1 and derives the pressure scale from the temperature scale. With rScale and pScale, we can now derive values for VelScale and BScale (see ''Data File Parameters" above). VelScale can now be used in conjunction with lScale to derive the TimeScale, which in turn allows us to obtain a value for ScaleCool.